A CLT Plancherel representations of the infinitedimensional unitary group
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Borodin, A., and A. Bufetov. “A CLT Plancherel representations
of the infinite-dimensional unitary group.” Journal of
Mathematical Sciences 190, no. 3 (April 20, 2013): 419-426.
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http://dx.doi.org/10.1007/s10958-013-1257-1
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arXiv:1203.3010v1 [math.RT] 14 Mar 2012
A CLT for Plancherel representations of the
infinite-dimensional unitary group
Alexei Borodin and Alexey Bufetov
Abstract
We study asymptotics of traces of (noncommutative) monomials formed by
images of certain elements of the universal enveloping algebra of the infinitedimensional unitary group in its Plancherel representations. We prove that they
converge to (commutative) moments of a Gaussian process that can be viewed as
a collection of simply yet nontrivially correlated two-dimensional Gaussian Free
Fields. The limiting process has previously arisen via the global scaling limit of
spectra for submatrices of Wigner Hermitian random matrices.
This note is an announcement, proofs will appear elsewhere.
1
Introduction
Asymptotic studies of measures on partitions of representation theoretic origin is
a well-known and popular subject. In addition to its intrinsic importance in representation theory, see e.g. [4] and references therein, it enjoys close connections
to the theory of random matrices, interacting particle systems, enumerative combinatorics, and other domains, for which it often provides crucial technical tools,
cf. e.g. [9], [13].
A typical scenario of how such measures arise is as follows: One starts with a
group with a well known list of irreducible representations, often parametrized by
partitions or related objects. Then a decomposition of a natural reducible representation of this group on irreducibles provides a split of the total dimension of the
representation space into dimensions of the corresponding isotypical components;
their relative sizes are the weights of the measure. This procedure is well-defined
for finite-dimensional representations, but also for infinite-dimensional representations with finite trace; the weight of (the label of) an isotypical component is then
defined as the trace of the projection operator onto it, provided that the trace is
normalized to be equal to 1 on the identity operator.
An alternative approach to measures of this sort consists in defining averages
with respect to such a measure for a suitable set of functions on labels of the irreducible representations. These averages are obtained as traces of the operators
in the ambient representation space that are scalar in each of the isotypical components. In their turn, the operators are images of central elements in the group
1
algebra of the group if the group is finite, or in the universal enveloping algebra of
the Lie algebra if one deals with a Lie group. The central elements form a commutative algebra that is being mapped to the algebra of functions on the labels,
i.e., on partitions or their relatives. The value of the function corresponding to
a central element at a representation label is the (scalar) value of this element in
that representation.
While one may be perfectly satisfied with such an approach from probabilistic
point of view, from representation theoretic point of view it is somewhat unsettling
that we are able to only deal with commutative subalgebras this way, while the
main interest of representation theory is in noncommutative effects.
The goal of this work is go beyond this commutativity constraint.
More exactly, in a specific setting of the finite trace representations of the
infinite-dimensional unitary group described below, we consider a family of commutative subalgebras of the universal enveloping algebra such that elements from
different subalgebras generally speaking do not commute. We further consider the
limit regime in which the measures for each of the commutative subalgebras are
known to approximate the two-dimensional Gaussian Free Field (GFF), see [3].
We want to study the “joint distribution” of these GFFs for different subalgebras,
whatever this might mean.
For any element of the universal enveloping algebra, one can define its “average”
as the trace of its image in the representation. Thus, having a representation, we
can define “averages” for arbitrary products of elements from our subalgebras,
despite the fact that the elements do not commute.
Our main result is that for certain Plancherel representations, these “averages”
converge to actual averages of suitable observables on a Gaussian process that
consists of a family of explicitly correlated GFFs. Thus, the original absence of
commutativity in this limit disappears, and yet the limiting GFFs that arise from
different commutative subalgebras do not become independent.
The same limiting object (the collection of correlated GFFs) has been previously shown to be the universal global scaling limit for eigenvalues of various
submatrices of Wigner Hermitian random matrices, cf. [2]. We also expect it to
arise from other, non-Plancherel factor representations of the infinite-dimensional
unitary group under appropriate limit transitions.
The present paper is an announcement, the proofs will appear in a subsequent
publication.
Acknowledgements
The authors are very grateful to Grigori Olshanski for numerous discussions that
were extremely helpful. A. Borodin was partially supported by NSF grant DMS1056390. A. Bufetov was partially supported by Simons Foundation-IUM scholarship, by Moebius Foundation for Young Scientists, and by RFBR–CNRS grant
10-01-93114.
2
2
Characters of unitary groups
Let I be a finite set of natural numbers, and let U (I) = (uij )i,j∈I be the group of
unitary matrices whose rows and columns are marked by elements of I. In what
follows we denote {1, 2, . . . , N } as 1, N . Consider the tower of embedded unitary
groups
U ({1}) ⊂ U ({1, 2}) ⊂ . . . U (1, N ) ⊂ U (1, N + 1) ⊂ . . . ,
where the embedding U (1, k) ⊂ U (1, k + 1) is defined by ui,k+1 = uk+1,i = 0,
1 ≤ i ≤ k, uk+1,k+1 = 1. The infinite–dimensional unitary group is the union of
these groups:
U (∞) =
∞
[
U (1, N ).
N =1
A signature (also called highest weight ) of length N is a sequence of N weakly
decreasing integers λ1 ≥ λ2 ≥ · · · ≥ λN . Let GTN denote the set of such signatures.
(Here the letters GT stand for ‘Gelfand-Tsetlin’.) We say that λ ∈ GTN and
µ ∈ GTN −1 interlace, notation µ ≺ λ, iff λi ≥ µi ≥ λi+1 for any 1 ≤ i ≤ N − 1.
We also define GT0 as a singleton consisting of an element that we denote as ∅.
We assume that ∅ ≺ λ for any λ ∈ GT1 .
S∞The Gelfand-Tsetlin graph GT is defined by specifying its set of vertices as
N =0 GTN and putting an edge between any two signatures λ and µ such that
either λ ≺ µ or µ ≺ λ. A path between signatures κ ∈ GTK and ν ∈ GTN , K < N
is a sequence
κ = λ(K) ≺ λ(K+1) ≺ · · · ≺ λ(N ) = ν,
λ(i) ∈ GTi .
Let DimN (ν) be the number of paths between ∅ and ν ∈ GTN . An infinite path
is a sequence
∅ ≺ λ(1) ≺ λ(2) ≺ · · · ≺ λ(k) ≺ λ(k+1) ≺ . . . .
We denote by P the set of all such paths. It is a topological space with the
topology
Q induced from the product topology on the ambient product of discrete
sets N ≥0 GTN .
For N = 0, 1, 2, . . . , let MN be a probability measure on GTN . We say that
{MN }∞
N =0 is a coherent system of measures if for any N ≥ 0 and λ ∈ GTN ,
MN (λ) =
X
MN +1 (ν)
ν:λ≺ν
DimN (λ)
.
DimN (ν)
Given a coherent system of measures {Mn }∞
n=1 , define a weight of a cylindric
set of P consisting of all paths with prescribed members up to GTN by
P (λ(1) , λ(2) , . . . , λ(N ) ) =
3
MN (λ(N ) )
.
DimN (λ(N ) )
(1)
Note that this weight depends on λ(N ) only. The coherency property implies that
these weights are consistent, and they correctly define a Borel probability measure
on P.
It is well known that the irreducible (complex) representations of U (N ) =
U (1, N ) can be parametrized by signatures of length N , and DimN (λ) is the dimension of the representation corresponding to λ. Let χλ be the conventional
character of this representation (i.e., the function on the group obtained by evaluating trace of the representation operators) divided by DimN (λ).
Define a character of the group U (∞) as a function χ : U (∞) → C that satisfies
1) χ(e) = 1, where e is the identity element of U (∞) (normalization);
2) χ(ghg −1 ) = χ(h), where g, h are any elements of U (∞) (centrality);
3) χ(gi gj−1 )ni,j=1 is an Hermitian and positive-definite matrix for any n ≥ 1 and
g1 , . . . , gn ∈ U (∞) (positive-definiteness);
4) the restriction of χ to U (1, N ) is a continuous function for any N ≥ 1
(continuity).
Let χ be a character of U (∞). It turns out that for any N ≥ 1, its restriction
to U (N ) can be decomposed into a series in χλ ,
X
MN (λ)χλ ,
χ|U(N ) =
λ∈GTN
and the coefficients MN (λ) form a coherent system of measures on GT. Conversely,
for any coherent system of measures on GT one can construct a character of U (∞)
using the above formula.
The space of characters of U (∞) is obviously convex. The extreme points of
this set can be viewed as traces of the factor-representations of U (∞) with finite
trace, or as spherical functions for irreducible spherical unitary representations of
the Gelfand pair (U (∞) × U (∞), diag(U (∞))), see [12] for details. The classification of the extreme characters is known as Edrei–Voiculescu theorem, see [10] and
references therein.
3 Characters and states on the universal enveloping algebra
Let gl(I) = (gij )i,j∈I be the complexified Lie algebra of U (I), let U(gl(I)) be its
universal enveloping algebra, and let Z(gl(I)) be the center of U(gl(I)). Denote by
[
U(gl(∞)) =
U(gl(1, N ))
N ≥1
the universal enveloping algebra of gl(∞).
There exists a canonical isomorphism
DI : U(gl(I)) → D(I),
4
where D(I) is the algebra of left-invariant differential operators on U (I) with complex coefficients. Let {xij } be the matrix coordinates. For any character χ define
a state on U(gl(∞)) as follows: For any X ∈ U(gl(∞))
hXiχ = DI (X)χ(xij )|xij =δij ,
X ∈ U(gl(I)).
(2)
Note that this definition is consistent for different choices of I.
We shall denote coordinates of the signatures that parametrize irreducible
(I)
(I)
representations of U (I) as λ1 , . . . , λ|I| . There exists a canonical isomorphism
Z(gl(I)) → A(I), where A(I) is the algebra of shifted symmetric polynomials in
(I)
(I)
λ1 , . . . , λ|I| , see e.g. [11]. For any central element, the value of the corresponding
function at a signature corresponds to the (scalar) operator that this element turns
into in the corresponding representation.
Similarly to Section 2, restricting χ to U (I) gives rise to a probability measure
on signatures of length |I|. One shows that the state h · iχ on an element of Z(gl(I))
equals the expectation of the corresponding function in A(I) with respect to this
probability measure.
One can also evaluate h · iχ as an expectation on a larger probability space.
Consider {1} ⊂ {1, 2} ⊂ . . . ⊂ 1, k ⊂ . . . . Let Z be the subalgebra in U(gl(∞))
generated by all the centers Z(gl(1, k)), k = 1, 2, . . . . To any element of Z(gl(1, k))
we assign a function on the path space P by taking the length k member of the
path and applying the isomorphism Z(gl(1, k)) → A(1, k). Hence, the algebra Z is
naturally embedded into functions on P. Denote the probability measure on P that
arises from the coherent system of measures originated from χ by µχ . Then the
value of h · iχ on any element of Z is equal to the expectation of the corresponding
function on the probability space (P, µχ ).
4
One-sided Plancherel characters
In what follows we restrict ourselves to the one–sided Plancherel character with a
growing parameter. This extreme character is defined as
!
∞
X
(xii − 1) ,
(3)
χ(U ) = exp γL
i=1
where U = [xij ]i,j≥1 , γ is fixed, and L is the growing parameter. Denote by µγ the
probability measure on P that corresponds to this character (cf. Section 2), and
denote by h · iγ the state that corresponds to this character (cf. Section 3).
If one restricts (3) to U (I) and decomposes it on normalized conventional characters of U (I), one obtains a probability measure on signatures of length |I| of the
form

λ1 +···+λ|I|

e−γL|I| (γL)
dim λ Dim|I| λ, λ1 ≥ · · · ≥ λ|I| ≥ 0;
γL
(λ1 + · · · + λ|I| )!
PI (λ) :=
(4)

0,
otherwise,
5
where dim λ is the dimension of the irreducible representation of the symmetric
group S|λ| that corresponds to λ (= the number of standard Young tableaux of
shape λ). Observe that these probability measures are supported by nonnegative
signatures, i.e., on partitions or Young diagrams.
γL
Asymptotic properties of P1,L
as L → ∞ and related distributions have been
extensively studied in [1], [3], [5], [7], [8].
5
Random height functions and GFF
A Gaussian family is a collection of Gaussian random variable {ξa }a∈Υ indexed by
an arbitrary set Υ. We assume that all the random variable are centered, i.e.
Eξa = 0,
for any a ∈ Υ.
Any Gaussian family gives rise to the covariance kernel Cov : Υ × Υ → R defined
(in the centered case) by
Cov(a1 , a2 ) = E(ξa1 ξa2 ).
Assume that a function C̃ : Υ × Υ → R is such that for any n ≥ 1 and
a1 , . . . , an ∈ Υ, [C̃(ai , aj )]ni,j=1 is a symmetric and positive-definite matrix. Then
(see e.g. [6]) there exists a centered Gaussian family with the covariance function
C̃.
Let H := {z ∈ C : I(z) > 0} be the upper half-plane, and let C0∞ be the space
of smooth real–valued compactly supported test functions on H. Define a function
C : C0∞ × C0∞ → R via
Z Z
z−w
1
dzdz̄dwdw̄.
ln
f1 (z)f2 (w) −
C(f1 , f2 ) :=
2π z − w̄
H H
The Gaussian Free Field (GFF) G on H with zero boundary conditions can
be defined as a Gaussian family {ξf }f ∈C0∞ with covariance kernel C. The field
G cannotRbe defined as a random function on H, but one can make sense of the
integrals f (z)G(z)dz over smooth finite contours in H with continuous functions
f (z), cf. [14].
Define the height function
H : R≥0 × R≥1 × P → N
as
H(x, y) =
(y)
o
√ n
(y)
π i ∈ 1, [y] : λi − i + 1/2 ≥ x ,
where λi are the coordinates of the signature of length [y] from the infinite path.
If we equip P with a probability measure µγ then H(x, y) becomes a random
function describing a certain random stepped surface, or a random lozenge tiling
of the half-plane, see [3].
6
Define x(z), y(z) : H → R via
x(z) = γ(1 − 2R(z)), y(z) = γ|z|2 .
Let us carry H(x, y) over to H — define
H Ω (z) = H(Lx(z), Ly(z)), z ∈ H.
It is known, cf. [1], [3], that there exists a limiting (nonrandom) height function
EH Ω (z)
,
L→∞
L
h̃(z) := lim
z ∈ H,
that describes the limit shape. The fluctuations around the limit shape were studied in [3], where it was shown that the fluctuation field
H(z) := H Ω (z) − EH Ω (z), z ∈ H,
(5)
converges to the GFF introduced above.
In [3], this convergence was proved for a certain space of test functions. Let
us formulate a similar statement that we prove in this work, and that utilizes a
different space of test functions.
Define a moment of the random height function via
Z ∞
xk H(Lx, Ly) − EH(Lx, Ly) dx.
My,k :=
−∞
Also define the corresponding moment of the GFF as
Z
dx(z)
My,k =
x(z)k G(z)
dz.
dz
2
z∈H;y=γ|z|
Proposition 1. As L → ∞, the collection of random variables {My,k }y>0,k∈Z≥0
converges, in the sense of finite dimensional distributions, to {My,k }y>0,k∈Z≥0 .
6
Convergence in the sense of states
Consider a probability space Ω and a sequence of k-dimensional random variables
(ηn1 , ηn2 , . . . , ηnk )n≥1 on it that converge, in the sense of convergence of moments, to
a Gaussian random vector (η 1 , . . . , η k ) with zero mean. If we define a state as
ξ ∈ L1 (Ω),
hξiΩ := Eξ,
then this convergence can be reformulated as
hηni1 ηni2 . . . ηnil iΩ −−−−→
n→∞
l/2
X Y
σ∈PM j=1
hη σ(2j−1) η σ(2j) iΩ ,
for any l ≥ 1 and any (i1 , . . . , il ) ∈ {1, . . . , k}l , (6)
7
where PM is the set of involutions on {1, 2, . . . , l} with no fixed points, also known
as perfect matchings. (In particular, PM is empty if l is odd). Indeed, Wick’s
formula implies that the right-hand side of (6) contains the moments of η.
Let A be a ∗-algebra and h · i be a state (=linear functional taking nonnegative
values at elements of the form aa∗ ) on it. Let a1 , a2 , . . . , ak ∈ A.
Assume that the elements a1 , . . . , ak and the state on A depend on a large
parameter L, and we also have a ∗-algebra A generated by elements a1 , . . . , ak and
a state φ on it. We say that (a1 , . . . , ak ) converge to (a1 , . . . , ak ) in the sense of
states if
hai1 ai2 . . . ail i −−−−→ φ(ai1 . . . ail ),
L→∞
(7)
and this holds for any l ∈ N and any index sets (i1 , i2 , . . . , il ) ∈ {1, 2, . . . , k}l .
We say that a collection {ai }i∈J ⊂ A indexed by an arbitrary set J and depending on a large parameter L, converges in the sense of states to a collection
{ai }i∈J ⊂ A if (7) holds for any finite subsets of {ai }i∈J and corresponding subsets
in {ai }i∈J .
In what follows, the algebra A is taken to be U(gl(∞)), and the state is taken
to be h · iγ , cf. (2), (3). The role of the limiting algebra A will be played by a
commutative algebra originated from a probability space.
7
The main result
Let A = {an }n≥1 be a sequence of pairwise distinct natural numbers. Let PA be
a copy of the path space P corresponding to A. Given A, we define the height
function
HA : R≥0 × R≥1 × PA → N
HA (x, y) =
({a1 ,...,a
})
o
√ n
({a1 ,...,a[y] })
π i ∈ 1, [y] : λi
− i + 1/2 ≥ x ,
[y]
where λi
are the coordinates of the length [y] signature in the infinite
path. Under the probability measure µγ on PA , HA (x, y) becomes a random function on the probability space (PA , µγ ).
Let {Ai }i∈J be a family of sequences of pairwise distinct natural numbers,
indexed by a set J. Introduce the notation
Ai = {ai,n }n≥1 ,
Ai,m = {ai,1 , . . . , ai,m }.
The coordinates ai,j = ai,j (L) may depend on the large parameter L.
We say that {Ai }i∈J is regular if for any i, j ∈ J and any x, y > 0 there exists
a limit
α(i, x; j, y) = lim
L→∞
8
|Ai,[xL] ∩ Aj,[yL] |
.
L
(8)
For example, the following family is regular: J = {1, 2, 3, 4} with a1,n = n,
a2,n = 2n, a3,n = 2n + 1, and


n = 1, 2, . . . , L,
n + L,
a4,n = n − L,
n = L + 1, L + 2, . . . , 2L,


n,
n ≥ 2L + 1.
Consider the union of copies of H indexed by J:
[
H(I) :=
Hi .
i∈I
Define a function C : H(J) × H(J) → R ∪ {+∞} via
α(i, |z|2 ; j, |w|2 ) − zw 1
, i, j ∈ J,
ln
Cij (z, w) =
2π α(i, |z|2 ; j, |w|2 ) − z w̄ z ∈ Hi , w ∈ Hj .
Proposition 2. For any regular family as above, there exists a generalized Gaussian process on H(J) with the covariance kernel Cij (z, w). More exactly, for any
finite set of test functions fm (z) ∈ C0∞ (Him ) and i1 , . . . , iM ∈ J, the covariance
matrix
Z Z
fk (z)fl (w)Cik il (z, w)dzdz̄dwdw̄
(9)
cov(fk , fl ) =
H
H
is positive-definite.
Proof. See [2, Proposition 1] .
Let us denote this Gaussian process as G{Ai }i∈J . Its restriction to a single
half-plane Hi is the GFF introduced above because
z − w
1
,
Cii (z, w) = −
ln z, w ∈ Hi , i ∈ J.
2π
z − w̄ As before, let us carry HA (x, y) over to H — define
Ω
HA
(z) = HA (Lx(z), Ly(z)), z ∈ H.
As was mentioned above, the fluctuations
Ω
Ω
(z),
(z) − EHA
Hi (z) := HA
i
i
i ∈ J, z ∈ Hi ,
(10)
for any fixed i converge to the GFF.
The main goal of this paper is to study the joint fluctuations (10) for different
i. The joint fluctuations of Hi are understood as follows. Define the moments of
the random height function as
Z ∞
(11)
Mi,y,k :=
xk HAi (Lx, Ly) − EHAi (Lx, Ly) dx.
−∞
9
It turns out that the function Mi,y,k belongs to A(Ai,[Ly] ), and thus it corresponds to an element of Z(gl(Ai,[Ly] )); denote this element by the same symbol.
Note that all such elements Mi,y,k for all i, y, k belong to the ambient algebra
U(gl(∞)), and we also have the state h · iγ defined on this ambient algebra. Thus,
we can talk about convergence of such elements in the sense of states, see Section
6. We are interested in the limit as L → ∞.
We prove that the family {Hi }i∈J converges to the generalized Gaussian process
G{Ai }i∈J . Define the moments of G{Ai }i∈J by
Mi,y,k =
Z
z∈H;y=γ|z|2
x(z)k GAi (z)
dx(z)
dz.
dz
Theorem 1. As L → ∞, the moments {Mi,y,k }i∈J,y>0,k∈Z≥0 converge in the sense
of states to the moments {Mi,y,k }i∈J,y>0,k∈Z≥0 .
Thus, in the L → ∞ limit, the noncommutativity disappears (limiting algebra
A is commutative), and yet the random fields Hi for different i’s are not independent.
Let u = Lx. The definition of the height function implies
[Ly]
√ X d
(A
)
δ u − λs i,[Ly] − s + 1/2 .
HAi (u, [Ly]) = − π
du
s=1
Define the shifted power sums
pk,I
k k !
|I|
X
1
1
(I)
λi − i +
,
=
− −i +
2
2
i=1
I ⊂ N.
One shows that pk,I ∈ A(I), and hence they correspond to certain elements of
Z(gl(I)) that we will denote by the same symbol.
Integrating (11) by parts shows that Mi,y,k can be rewritten as


k+1
k+1
√
[Ly] [Ly] X
X
L
π
1
1
(A
)
(A
)

λs i,[Ly] − s +
λs i,[Ly] − s +
−E
k+1
2
2
s=1
s=1
√
L−(k+1) π
(pk+1,I − Epk+1,I ) .
=
k+1
−(k+1)
Thus, Theorem 1 can be reformulated as follows.
Theorem 2. Let k1 , . . . , km ≥ 1 and I1 , . . . , Im be finite subsets of N that may
depend on the large parameter L in such a way that there exist limits
ηr = lim
L→∞
|Ir |
> 0,
L
crs = lim
L→∞
10
|Ir ∩ Is |
.
L
Then as L → ∞, the collection
L−kr (pkr ,Ir − Epkr ,Ir )
m
r=1
of elements of U(gl(∞)) converges in the sense of states, cf. (7), to the Gaussian
vector (ξ1 , . . . , ξm ) with zero mean and covariance
I
I
kr ks
Eξr ξs =
(x(z))kr −1 (x(w))ks −1
π
|z|2 = ηγr ;I(z)>0 |w|2 = ηγs ;I(w)>0
crs /γ − zw d(x(z)) d(x(w))
1
×
ln dzdw.
2π
crs /γ − z w̄ dz
dw
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